Bühlmann decompression algorithm

The Bühlmann decompression algorithm is a mathematical model (algorithm) of the way in which inert gases enter and leave the human body as the ambient pressure changes.^[1] Versions are used to create Bühlmann decompression tables and in personal dive computers to compute no-decompression limits and decompression schedules for dives in real-time. These decompression tables allow divers to plan the depth and duration for dives and the required decompression stops.

The algorithm was developed by Swiss physician Dr. Albert A. Bühlmann, who did research into decompression theory at the Laboratory of Hyperbaric Physiology at the University Hospital in Zürich, Switzerland.^[2]^[3] The results of Bühlmann's research that began in 1959 were published in a 1983 German book whose English translation was entitled Decompression-Decompression Sickness.^[1] The book was regarded as the most complete public reference on decompression calculations and was used soon after in dive computer algorithms.

The model assumes perfusion limited gas exchange and multiple parallel tissue compartments and uses an inverse exponential model for in-gassing and out-gassing, both of which are assumed to occur in the dissolved phase (without bubble formation).

Principles

Building on the previous work of John Scott Haldane and Robert Workman,^[4]^[5] and working off funding from Shell Oil Company,^[6] Bühlmann designed studies to establish the longest half-times of nitrogen and helium in human tissues.^[1] These studies were confirmed by the Capshell experiments in the Mediterranean Sea in 1966.^[6]^[7]

The basic idea is to represent the human body by multiple tissues (compartments) of different saturation half-times and to calculate the partial pressure $P$ of the inert gases in each of the $n$ compartments:

$P=P_{0}+(P_{gas}-P_{0})\cdot (1-2^{-{\frac {t_{exp}}{t_{1/2}}}})$

with the initial partial pressure $P_{0}$ , the partial pressure in the breathing gas $P_{gas}$ (minus the vapour pressure of water in the lung of about 60 mbar), the time of exposure $t_{exp}$ and the compartment-specific saturation half-time $t_{1/2}$ .

When the gas pressure drops, the compartments start to off-gas. To calculate the minimum tolerable pressure $P_{tol}$ , the constants $a$ and $b$ , which are derived from the saturation half-time as follows (ZH-L16A):

$a={\frac {2\,{\text{atm}}}{\sqrt[{3}]{\frac {t_{1/2}}{\text{min}}}}}$

$b=1.005-{\frac {1}{\sqrt[{2}]{\frac {t_{1/2}}{\text{min}}}}}$

are used:

$P_{tol}=(P-a)\cdot b$

Different versions calculate $a$ and $b$ differently, use different half-times or fewer compartments.

Table of ZH-L16A Half-times $t_{1/2}$ with $a$ and $b$ values for nitrogen (N₂) and helium (He).^[8]^[9]
Compartment	$t_{1/2,{\text{N}}_{2}}$ / min	$a_{{\text{N}}_{2}}$	$b_{{\text{N}}_{2}}$	$t_{1/2,{\text{He}}}$ / min	$a_{\text{He}}$	$b_{\text{He}}$
01	004	1.2599	0.5050	001.51	1.7474	0.4245
02	008	1.0000	0.6514	003.02	1.3830	0.5747
03	012.5	0.8618	0.7222	004.72	1.1919	0.6527
04	018.5	0.7562	0.7825	006.99	1.0458	0.7223
05	027	0.6667	0.8125	010.21	0.9220	0.7582
06	038.3	0.5933	0.8434	014.48	0.8205	0.7957
07	054.3	0.5282	0.8693	020.53	0.7305	0.8279
08	077	0.4701	0.8910	029.11	0.6502	0.8553
09	109	0.4187	0.9092	041.2	0.5950	0.8757
10	146	0.3798	0.9222	055.19	0.5545	0.8903
11	187	0.3497	0.9319	070.69	0.5333	0.8997
12	239	0.3223	0.9403	090.34	0.5189	0.9073
13	305	0.2971	0.9477	115.29	0.5181	0.9122
14	390	0.2737	0.9544	147.42	0.5176	0.9171
15	498	0.2523	0.9602	188.24	0.5172	0.9217
16	635	0.2327	0.9653	240.03	0.5119	0.9267

Versions

Several versions of the Bühlmann algorithm have been developed, both by Bühlmann and by later workers. The naming convention used to identify the algorithms is a code starting ZH-L, from Zürich (ZH), limits (L) followed by the number of tissue compartments, and other unique identifiers. For example:

ZHL-16 or ZH-L16A: The original 16-compartment algorithm (no conservatism at all).
ZHL-16B: The 16-compartment algorithm modified for dive table production, using slightly more conservative “a” values, mainly in the middle compartments.^[8] Recently used in dive computers with high performance processor units, it is more flexible (especially in tech dives) compared to the ZHL16C
ZHL-16C: The 16-compartment algorithm with further modification to the middle and faster “a” values, intended for use in dive computers as a "package". It can be used with almost all low-level processor units but it is less flexible compared to the ZHL16B.
ZHL-16 ADT DD: 16-compartment adaptive model used by Uwatec for their trimix-enabled computers. Modified in the middle compartments from the original ZHL-C, is adaptive to diver workload and includes Profile-Determined Intermediate Stops. Profile modification is by means of "MB Levels", personal option conservatism settings, which are not defined in the manual.^[10]
ZHL-12: ^[11]
ZHL-8: A version using a reduced number of tissue compartments to reduce the computational load for personal dive computers.
ZHL-8 ADT: 8-compartment adaptive model used by Uwatec. This model may reduce the no-stop limit or require the diver to complete a compensatory decompression stop after an ascent rate violation, high work level during the dive, or low water temperature. This algorithm is used in computers which can accurately monitor air consumption and instantaneous rate of air consumption to model work load (exertion) via changes in the rate of gas consumption, which allows plausible modelling of additional decompression obligation based on exertion at depth. It also monitors ambient temperature and selects the choice of risk tissue accordingly. This results in earlier and longer decompression requirements in colder water.^[12]^[13]
ZHL-8 ADT MB: A version of the ZHL-8 ADT claimed to suppress MicroBubble formation.^[12]
ZHL-8 ADT MB PDIS: Profile-Determined Intermediate Stops.^[14]
ZHL-8 ADT MB PMG: Predictive Multi-Gas.^{[citation needed]}

Ascent rates

Ascent rate is intrinsically a variable, and may be selected by the programmer or user for table generation or simulations, and measured as real-time input in dive computer applications.

The rate of ascent to the first stop is limited to 3 bar per minute for compartments 1 to 5, 2 bar per minute for compartments 6 and 7, and 1 bar per minute for compartments 8 to 16. Chamber decompression may be continuous, or if stops are preferred they may be done at intervals of 1 or 3 m.^[11]

Gradient factors

User input of gradient factors is sometimes available for planning and real time applications.^[15]

Tables

Max Hahn first used Bühlmann's algorithm to develop dive tables for the Swiss Underwater Sport Association. In 1987, the SAA Bühlmann System was developed by Bob Cole. This system used the dive tables and a set of rules so that people could dive safely and stay below their no-decompression limit.^[16] The tables are still used today and are very popular; many dive computers still use the ZHL-8 algorithm and many tables are based on the ZHL-16 algorithm or derivatives. These calculations also include considerations for repetitive and altitude diving.^[1]^[17]^[18]^[19]^[20]^[21]

References

^ ^a ^b ^c ^d Bühlmann, Albert A. (1984). Decompression-Decompression Sickness. Berlin New York: Springer-Verlag. ISBN 0-387-13308-9.
^ Bühlmann, Albert A. (1982). "[Experimental principles of risk-free decompression following hyperbaric exposure. 20 years of applied decompression research in Zurich]". Schweizerische Medizinische Wochenschrift (in German). 112 (2): 48–59. PMID 7071573.
^ Wendling, J; Nussberger, P; Schenk, B (1999). "Milestones of the deep diving research laboratory Zurich". South Pacific Underwater Medicine Society Journal. 29 (2). ISSN 0813-1988. OCLC 16986801. Archived from the original on 2012-02-03. Retrieved 2009-04-02.{{cite journal}}: CS1 maint: unfit URL (link)
^ Boycott, A.E.; Damant, G.C.C.; Haldane, John Scott (1908). "Prevention of compressed air illness". Journal of Hygiene. Cambridge University Press. 8 (3): 342–443. doi:10.1017/S0022172400003399. PMC 2167126. PMID 20474365. Archived from the original on 2011-03-24. Retrieved 2009-06-12.{{cite journal}}: CS1 maint: unfit URL (link)
^ Workman, Robert D. (1957). "Calculation of air saturation decompression tables". Navy Experimental Diving Unit Technical Report. NEDU-RR-11-57. Archived from the original on 2011-09-18. Retrieved 2009-06-12.{{cite journal}}: CS1 maint: unfit URL (link)
^ ^a ^b Völlm, Ernst B.; Götte, Georg (1994). "Leading diving researcher dies unexpectedly: Albert A Bühlmann, 1923 - 1994". Pressure, Newsletter of the Undersea and Hyperbaric Medical Society. 23 (3): 1–3. ISSN 0889-0242.
^ Bühlmann, Albert A.; Frei, P.; Keller, Hannes (October 1967). "Saturation and desaturation with N2 and He at 4 atm". Journal of Applied Physiology. 23 (4): 458–62. doi:10.1152/jappl.1967.23.4.458. PMID 6053671.
^ ^a ^b Chapman, Paul. "An explanation Professor A A Buhlmann's ZH-L16 Algorithm" (PDF). Retrieved 28 July 2022 – via Pambucol, Vlad, (25 June 2002) "Dive Computers", Annex 1, pages 46-54, DiveCon project 6/25/02.
^ Bühlmann, A.A.; Völlm, E.B.; Nussberger, P. (2002). Tauchmedizin (in German). Springer-Verlag. doi:10.1007/978-3-642-55939-6. ISBN 978-3-642-55939-6.
^ Technical diving software for Galilio: User manual (PDF). Scubapro. Retrieved 18 September 2019.
^ ^a ^b Bühlmann, A.A. (1984). Decompression - Decompression Sickness. Springer -Verlag. doi:10.1007/978-3-662-02409-6. ISBN 978-3-662-02409-6.
^ ^a ^b Staff. "Smart microbubble management" (PDF). In Depth. Uwatec. Retrieved 12 March 2016.
^ Marroni, A; Cali Corleo, R; Balestra, C; Longobardi, P; Voellm, E; Pieri, M; Pepoli, R (2000). "Effects of the variation of Ascent Speed and Profile on the production of Circulating Venous Gas Emboli and the Incidence of DCI in Compressed Air Diving. Phase 1. Introduction of extra deep stops in the ascent profile without changing the original ascent rates. DSL Special Project 01/2000" (PDF). Divers Alert Network. Retrieved 12 March 2016.
^ Staff. "Diving with PDIS (Profile-Dependent Intermediate Stop)" (PDF). Dykkercentret website. Frederiksberg: Dykkercentret ApS. Archived from the original (PDF) on 17 October 2016. Retrieved 5 March 2016.
^ "Perdix AI operating instructions" (PDF). Shearwater. Retrieved 10 October 2019.
^ Powell, Mark (2008). Deco for Divers. Southend-on-Sea: Aquapress. p. 18. ISBN 978-1-905492-07-7.
^ Bühlmann, Albert A. (1987). "Decompression after repeated dives". Undersea Biomedical Research. 14 (1): 59–66. ISSN 0093-5387. OCLC 2068005. PMID 3810993. Archived from the original on July 7, 2012. Retrieved 2008-04-25.{{cite journal}}: CS1 maint: unfit URL (link)
^ Böni, M; Schibli, R; Nussberger, P; Bühlmann, Albert A (1976). "Diving at diminished atmospheric pressure: air decompression tables for different altitudes". Undersea Biomedical Research. 3 (3): 189–204. ISSN 0093-5387. OCLC 2068005. PMID 969023. Archived from the original on July 8, 2012. Retrieved 2008-04-24.{{cite journal}}: CS1 maint: unfit URL (link)
^ Bühlmann, Albert A.; Schibli, R.; Gehring, H. (March 1973). "[Experimental studies on decompression following diving in mountain lakes at reduced air pressure]". Schweizerische Medizinische Wochenschrift (in German). 103 (10): 378–83. PMID 4144210.
^ Bühlmann, Albert A. (1989). "[Decompression problems in diving in mountain lakes]". Schweizerische Zeitschrift für Sportmedizin (in French). 37 (2): 80–3, discussion 99–102. PMID 2799365.
^ Bühlmann, Albert A. (1984). "[Decompression during lowered air pressure]". Schweizerische Medizinische Wochenschrift (in German). 114 (26): 942–7. PMID 6087447.

External links

Many articles on the Bühlmann tables are available on the web.

Chapman, Paul (November 1999). "An Explanation of Professor A.A. Buehlmann's ZH-L16 Algorithm". New Jersey Scuba Diver. Archived from the original on 2010-02-15. Retrieved 20 January 2010. – Detailed background and worked examples
Decompression Theory: Robert Workman and Prof A Bühlmann. An overview of the history of Bühlmann tables
Stuart Morrison: DIY Decompression (2000). Works through the steps involved in using Bühlmann's ZH-L16 algorithm to write a decompression program.

[AAB1984-1] Bühlmann, Albert A. (1984). Decompression-Decompression Sickness. Berlin New York: Springer-Verlag. ISBN 0-387-13308-9.

[AAB1982-2] Bühlmann, Albert A. (1982). "[Experimental principles of risk-free decompression following hyperbaric exposure. 20 years of applied decompression research in Zurich]". Schweizerische Medizinische Wochenschrift (in German). 112 (2): 48–59. PMID 7071573.

[spums1999-3] Wendling, J; Nussberger, P; Schenk, B (1999). "Milestones of the deep diving research laboratory Zurich". South Pacific Underwater Medicine Society Journal. 29 (2). ISSN 0813-1988. OCLC 16986801. Archived from the original on 2012-02-03. Retrieved 2009-04-02.{{cite journal}}: CS1 maint: unfit URL (link)

[haldane-4] Boycott, A.E.; Damant, G.C.C.; Haldane, John Scott (1908). "Prevention of compressed air illness". Journal of Hygiene. Cambridge University Press. 8 (3): 342–443. doi:10.1017/S0022172400003399. PMC 2167126. PMID 20474365. Archived from the original on 2011-03-24. Retrieved 2009-06-12.{{cite journal}}: CS1 maint: unfit URL (link)

[Workman57-5] Workman, Robert D. (1957). "Calculation of air saturation decompression tables". Navy Experimental Diving Unit Technical Report. NEDU-RR-11-57. Archived from the original on 2011-09-18. Retrieved 2009-06-12.{{cite journal}}: CS1 maint: unfit URL (link)

[Pressure94-6] Völlm, Ernst B.; Götte, Georg (1994). "Leading diving researcher dies unexpectedly: Albert A Bühlmann, 1923 - 1994". Pressure, Newsletter of the Undersea and Hyperbaric Medical Society. 23 (3): 1–3. ISSN 0889-0242.

[pmid6053671-7] Bühlmann, Albert A.; Frei, P.; Keller, Hannes (October 1967). "Saturation and desaturation with N2 and He at 4 atm". Journal of Applied Physiology. 23 (4): 458–62. doi:10.1152/jappl.1967.23.4.458. PMID 6053671.

[Chapman-8] Chapman, Paul. "An explanation Professor A A Buhlmann's ZH-L16 Algorithm" (PDF). Retrieved 28 July 2022 – via Pambucol, Vlad, (25 June 2002) "Dive Computers", Annex 1, pages 46-54, DiveCon project 6/25/02.

[Buhlmann_et_al_2002-9] Bühlmann, A.A.; Völlm, E.B.; Nussberger, P. (2002). Tauchmedizin (in German). Springer-Verlag. doi:10.1007/978-3-642-55939-6. ISBN 978-3-642-55939-6.

[Galileo-10] Technical diving software for Galilio: User manual (PDF). Scubapro. Retrieved 18 September 2019.

[Buhlmann_1984-11] Bühlmann, A.A. (1984). Decompression - Decompression Sickness. Springer -Verlag. doi:10.1007/978-3-662-02409-6. ISBN 978-3-662-02409-6.

[Smart-12] Staff. "Smart microbubble management" (PDF). In Depth. Uwatec. Retrieved 12 March 2016.

[13] Marroni, A; Cali Corleo, R; Balestra, C; Longobardi, P; Voellm, E; Pieri, M; Pepoli, R (2000). "Effects of the variation of Ascent Speed and Profile on the production of Circulating Venous Gas Emboli and the Incidence of DCI in Compressed Air Diving. Phase 1. Introduction of extra deep stops in the ascent profile without changing the original ascent rates. DSL Special Project 01/2000" (PDF). Divers Alert Network. Retrieved 12 March 2016.

[dykcen-14] Staff. "Diving with PDIS (Profile-Dependent Intermediate Stop)" (PDF). Dykkercentret website. Frederiksberg: Dykkercentret ApS. Archived from the original (PDF) on 17 October 2016. Retrieved 5 March 2016.

[Perdix_AI-15] "Perdix AI operating instructions" (PDF). Shearwater. Retrieved 10 October 2019.

[Powell2008-18-16] Powell, Mark (2008). Deco for Divers. Southend-on-Sea: Aquapress. p. 18. ISBN 978-1-905492-07-7.

[AAB1987-17] Bühlmann, Albert A. (1987). "Decompression after repeated dives". Undersea Biomedical Research. 14 (1): 59–66. ISSN 0093-5387. OCLC 2068005. PMID 3810993. Archived from the original on July 7, 2012. Retrieved 2008-04-25.{{cite journal}}: CS1 maint: unfit URL (link)

[Boni1976-18] Böni, M; Schibli, R; Nussberger, P; Bühlmann, Albert A (1976). "Diving at diminished atmospheric pressure: air decompression tables for different altitudes". Undersea Biomedical Research. 3 (3): 189–204. ISSN 0093-5387. OCLC 2068005. PMID 969023. Archived from the original on July 8, 2012. Retrieved 2008-04-24.{{cite journal}}: CS1 maint: unfit URL (link)

[AAB1973-19] Bühlmann, Albert A.; Schibli, R.; Gehring, H. (March 1973). "[Experimental studies on decompression following diving in mountain lakes at reduced air pressure]". Schweizerische Medizinische Wochenschrift (in German). 103 (10): 378–83. PMID 4144210.

[AAB1989-20] Bühlmann, Albert A. (1989). "[Decompression problems in diving in mountain lakes]". Schweizerische Zeitschrift für Sportmedizin (in French). 37 (2): 80–3, discussion 99–102. PMID 2799365.

[AAB1984b-21] Bühlmann, Albert A. (1984). "[Decompression during lowered air pressure]". Schweizerische Medizinische Wochenschrift (in German). 114 (26): 942–7. PMID 6087447.

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